Can a character take free action out of her turn?

Free actionsD20PFSRD in Pathfinder take almost no time. But can they be performed outside of one’s turn or in response to some trigger?

Scenario 1: Can a Barbarian enter rage in response to taking damage outside of her turn? Likewise, can anyone drop prone in response to being attacked from distance?

Scenario 2: Can a Barbarian enter rage in response to taking damage during her turn (for example, as a result of an attack of opportunity)?